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<h1 class="heading"><a href="MATH-2023-OPDE.html"><span class="title">MATH 2023: Ordinary and Partial Differential Equations</span></a></h1>
<p class="byline">Xiaoyi Chen and Wei Zhang</p>
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<a href="ch_first.html" data-scroll="ch_first" class="internal"><span class="codenumber">1</span> <span class="title">Introduction</span></a><ul>
<li><a href="sec_1-intro.html" data-scroll="sec_1-intro" class="internal">Classification of Differential Equations</a></li>
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<a href="ch_second.html" data-scroll="ch_second" class="internal"><span class="codenumber">2</span> <span class="title">First Order Ordinary Differential Equations</span></a><ul>
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<li><a href="sec2_2.html" data-scroll="sec2_2" class="internal">Further Discussion of Linear Equations (For reading only)</a></li>
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<li><a href="sec3_2.html" data-scroll="sec3_2" class="internal">Fundamental Solutions of Linear Homogeneous Equations</a></li>
<li><a href="sec3_3.html" data-scroll="sec3_3" class="internal">Linear Independence and Wronskian</a></li>
<li><a href="sec3_4.html" data-scroll="sec3_4" class="internal">Complex roots of the characteristic equations</a></li>
<li><a href="sec3_5.html" data-scroll="sec3_5" class="internal">Repeated Roots: Reduction of Order</a></li>
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<li><a href="sec4_2.html" data-scroll="sec4_2" class="internal">Homogeneous Equations with Constant Coefficients</a></li>
<li><a href="sec4_3.html" data-scroll="sec4_3" class="internal">The Method of Undetermined Coefficients</a></li>
<li><a href="sec4_4.html" data-scroll="sec4_4" class="internal">The Method of Variation of Parameters</a></li>
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<li><a href="sec5_1.html" data-scroll="sec5_1" class="internal">Brief Review on Power Series</a></li>
<li><a href="sec5_2.html" data-scroll="sec5_2" class="internal">Introduction</a></li>
<li><a href="sec5_3.html" data-scroll="sec5_3" class="internal">Series Solutions Near an Ordinary Point</a></li>
<li><a href="sec5_4.html" data-scroll="sec5_4" class="internal">Euler’s Equation</a></li>
<li><a href="sec5_5.html" data-scroll="sec5_5" class="internal">Series Solution near a Regular Singular Point</a></li>
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<a href="ch_six.html" data-scroll="ch_six" class="internal"><span class="codenumber">6</span> <span class="title">System of First Order Linear Equations</span></a><ul>
<li><a href="sec6_1.html" data-scroll="sec6_1" class="internal">Introduction <span class="process-math">\(\&amp;\)</span> Basic Theory</a></li>
<li><a href="sec6_2.html" data-scroll="sec6_2" class="internal">Homogeneous System with Constant Coefficients</a></li>
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<a href="ch_seven.html" data-scroll="ch_seven" class="internal"><span class="codenumber">7</span> <span class="title">Partial Differential Equations</span></a><ul>
<li><a href="sec7_1.html" data-scroll="sec7_1" class="internal">Two-Point Boundary Value Problems</a></li>
<li><a href="sec7_2.html" data-scroll="sec7_2" class="internal">Eigenvalue Problems</a></li>
<li><a href="sec7_3.html" data-scroll="sec7_3" class="internal">Fourier Series</a></li>
<li><a href="sec7_4.html" data-scroll="sec7_4" class="internal">The Fourier Convergence Theorem</a></li>
<li><a href="sec7_5.html" data-scroll="sec7_5" class="internal">Even and Odd Functions</a></li>
<li><a href="sec7_6.html" data-scroll="sec7_6" class="internal">Introduction to Partial Differential Equations</a></li>
<li><a href="sec7_7.html" data-scroll="sec7_7" class="active">1D Heat Equation; Solutions by Separation of Variable and Fourier Series</a></li>
<li><a href="sec7_8.html" data-scroll="sec7_8" class="internal">Other Heat Conduction Problems</a></li>
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<a href="ch_eight.html" data-scroll="ch_eight" class="internal"><span class="codenumber">8</span> <span class="title">Laplace transform</span></a><ul>
<li><a href="sec8_1.html" data-scroll="sec8_1" class="internal">What are Laplace Transforms, and Why?</a></li>
<li><a href="sec8_2.html" data-scroll="sec8_2" class="internal">Finding Laplace Transforms</a></li>
<li><a href="sec8_3.html" data-scroll="sec8_3" class="internal">Finding inverse transforms using partial fractions</a></li>
<li><a href="sec8_4.html" data-scroll="sec8_4" class="internal">Solving ODEs and ODE Systems</a></li>
<li><a href="sec8_5.html" data-scroll="sec8_5" class="internal">Step input and Impulse problems</a></li>
<li><a href="sec8_6.html" data-scroll="sec8_6" class="internal">Laplace transform for PDE (heat equation)</a></li>
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<a href="ch_features.html" data-scroll="ch_features" class="internal"><span class="codenumber">9</span> <span class="title">Examples of PreTeXt features</span></a><ul><li><a href="sec_features-blocks.html" data-scroll="sec_features-blocks" class="internal">Environments and Blocks</a></li></ul>
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<li class="link"><a href="solutions-1.html" data-scroll="solutions-1" class="internal"><span class="codenumber">A</span> <span class="title">Selected Hints</span></a></li>
<li class="link"><a href="solutions-2.html" data-scroll="solutions-2" class="internal"><span class="codenumber">B</span> <span class="title">Selected Solutions</span></a></li>
<li class="link"><a href="appendix-1.html" data-scroll="appendix-1" class="internal"><span class="codenumber">C</span> <span class="title">List of Symbols</span></a></li>
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<main class="main"><div id="content" class="pretext-content"><section class="section" id="sec7_7"><h2 class="heading hide-type">
<span class="type">Section</span> <span class="codenumber">7.7</span> <span class="title">1D Heat Equation; Solutions by Separation of Variable and Fourier Series</span>
</h2>
<p id="p-401">Consider a heat conduction problem for a straight bar of uniform cross section and homogeneous material. Let the <span class="process-math">\(x\)</span>-axis be chosen to lie along the axis of the bar, and let <span class="process-math">\(x = 0\)</span> and <span class="process-math">\(x = L\)</span> denote the ends of the bar. Suppose further that the sides of the bar are perfectly insulated so that no heat passes through them and the ends of the bar are held at fixed temperature <span class="process-math">\(0\text{.}\)</span> We also assume that the cross-sectional dimensions are so small that the temperature <span class="process-math">\(u\)</span> can be considered as constant on any given cross section. Then <span class="process-math">\(u\)</span> is a function only of the axial coordinate <span class="process-math">\(x\)</span> and the time <span class="process-math">\(t\text{.}\)</span></p>
<p id="p-402">The variation of temperature in the bar is governed by a partial differential equation. The equation is called the <dfn class="terminology">heat conduction equation</dfn>, and has the form</p>
<div class="displaymath process-math" data-contains-math-knowls="  ">
\begin{equation*}
u_t=\alpha^2u_{xx}\label{heatpde},\quad 0&lt;x&lt;L,\quad t&gt;0,
\end{equation*}
</div>
<p class="continuation">where <span class="process-math">\(\alpha^2\)</span> is a constant known as the thermal diffusivity. In addition, we assume that the initial temperature distribution in the bar is given; thus</p>
<div class="displaymath process-math" data-contains-math-knowls="  ">
\begin{equation*}
u(x,0)=f(x),\quad 0\leq x\leq L,\label{heatic}
\end{equation*}
</div>
<p class="continuation">where <span class="process-math">\(f\)</span> is a given function. Finally, we assume that the ends of the bar are held at fixed temperature zero:</p>
<div class="displaymath process-math" data-contains-math-knowls="  ">
\begin{equation*}
u(0,t)=0,\quad u(L,t)=0,\quad t&gt;0.\label{heatbc}
\end{equation*}
</div>
<p class="continuation">The fundamental problem of heat conduction is to find <span class="process-math">\(u(x, t)\)</span> that satisfies the <dfn class="terminology">boundary-value problem</dfn> (BVP), that is the partial differential equation <code class="code-inline tex2jax_ignore">[cross-reference to target(s) "heatpde" missing or not unique]</code> together with the initial condition <code class="code-inline tex2jax_ignore">[cross-reference to target(s) "heatic" missing or not unique]</code> and boundary conditions <code class="code-inline tex2jax_ignore">[cross-reference to target(s) "heatbc" missing or not unique]</code>.</p>
<p id="p-403"><dfn class="terminology">Step 1: Separating variables.</dfn> To find <span class="process-math">\(u(x,t)\text{,}\)</span> we start by making a basic assumption about the form of the solutions: <span class="process-math">\(u(x,t)\)</span> is a product of two functions, one depending only on <span class="process-math">\(x\)</span> and the other depending only on <span class="process-math">\(t\text{;}\)</span> thus</p>
<div class="displaymath process-math" data-contains-math-knowls="        ">
\begin{equation*}
u(x,t)=X(x)T(t).\label{separation}
\end{equation*}
</div>
<p class="continuation">Substituting from Eq. <code class="code-inline tex2jax_ignore">[cross-reference to target(s) "separation" missing or not unique]</code> for <span class="process-math">\(u\)</span> in the differential equation <code class="code-inline tex2jax_ignore">[cross-reference to target(s) "heatpde" missing or not unique]</code> yields</p>
<div class="displaymath process-math" data-contains-math-knowls="        ">
\begin{equation*}
XT'=\alpha^2X''T,\quad\to\quad \frac{X''}{X}=\frac{1}{\alpha^2}\frac{T'}{T},\label{sepa}
\end{equation*}
</div>
<p class="continuation">in which the variables are separated; that is, the left side depends only on <span class="process-math">\(x\)</span> and the right side only on <span class="process-math">\(t\text{.}\)</span> The only way to make it happen is the radio equals a constant. If we call this separation constant <span class="process-math">\(-\lambda\text{,}\)</span> then</p>
<div class="displaymath process-math" data-contains-math-knowls="        ">
\begin{equation*}
\frac{X''}{X}=\frac{1}{\alpha^2}\frac{T'}{T}=-\lambda,
\end{equation*}
</div>
<p class="continuation">We end up with 2 ODEs,</p>
<div class="displaymath process-math" data-contains-math-knowls="        ">
\begin{equation*}
\begin{aligned}
X''+\lambda X &amp;=&amp; 0\label{ode1}\\
T' + \alpha^2\lambda T &amp;=&amp; 0 \label{ode2}
\end{aligned}
\end{equation*}
</div>
<p class="continuation">The assumption <code class="code-inline tex2jax_ignore">[cross-reference to target(s) "separation" missing or not unique]</code> has led to the replacement of the partial differential equation <code class="code-inline tex2jax_ignore">[cross-reference to target(s) "heatpde" missing or not unique]</code> by two ODEs <code class="code-inline tex2jax_ignore">[cross-reference to target(s) "ode1" missing or not unique]</code> and <code class="code-inline tex2jax_ignore">[cross-reference to target(s) "ode2" missing or not unique]</code>. Each of these equations can be readily solved for any value of <span class="process-math">\(\lambda\text{.}\)</span> The product of two solutions of Eq. <code class="code-inline tex2jax_ignore">[cross-reference to target(s) "ode1" missing or not unique]</code> and <code class="code-inline tex2jax_ignore">[cross-reference to target(s) "ode2" missing or not unique]</code>, respectively, provides a solution of the partial differential equation <code class="code-inline tex2jax_ignore">[cross-reference to target(s) "heatpde" missing or not unique]</code>.</p>
<p id="p-404">However, we are interested only in those solutions of Eq. <code class="code-inline tex2jax_ignore">[cross-reference to target(s) "heatpde" missing or not unique]</code> that also satisfy the boundary conditions <code class="code-inline tex2jax_ignore">[cross-reference to target(s) "heatbc" missing or not unique]</code>.</p>
<p id="p-405">Substituting <span class="process-math">\(u(x, t)=XT\)</span> in the boundary condition at <span class="process-math">\(x = 0\text{,}\)</span> we obtain</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
u(0,t)=X(0)T(t)=0.
\end{equation*}
</div>
<p class="continuation">If above equation is satisfied by choosing <span class="process-math">\(T(t)\)</span> to be zero for all <span class="process-math">\(t\text{,}\)</span> then <span class="process-math">\(u(x,t)\)</span> is zero for all <span class="process-math">\(x\)</span> and <span class="process-math">\(t\text{,}\)</span> and we have already rejected this possibility. Therefore it must be satisfied by requiring that</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
X(0)=0.
\end{equation*}
</div>
<p class="continuation">Similarly, the boundary condition at <span class="process-math">\(x = L\)</span> requires that</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
X(L)=0.
\end{equation*}
</div>
<p class="continuation">We have the following <em class="emphasis">eigenvalue problem</em> for <span class="process-math">\(X(x)\)</span></p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
X''+\lambda X=0,\quad X(0)=X(L)=0,
\end{equation*}
</div>
<p class="continuation">which is an example we had earlier. The nontrivial solutions should be eigenfunctions</p>
<div class="displaymath process-math" data-contains-math-knowls="" id="Xsol">
\begin{equation}
{\lambda_n}={\left(\frac{n\pi}{L}\right)^2},\quad {X_n(x)}={\sin\frac{n\pi x}{L}},\quad n={1},2,3,\cdots\smallskip\tag{7.7.1}
\end{equation}
</div>
<p id="p-406">For a given <span class="process-math">\(n\text{,}\)</span> we get a solution <span class="process-math">\(T_n(t)\text{,}\)</span> which solves</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
\begin{aligned}
&amp;T'(t)+\alpha^2\lambda_n T ~~= ~~T'(t)+ \alpha^2 \frac{n^2\pi^2}{L^2} T~~ = ~0&amp;\nonumber\\
&amp;T_n(t)=C_n\cdot\exp\left[-\left(\frac{n\pi\alpha}{L}\right)^2t\right],&amp;\label{Tsol}
\end{aligned}
\end{equation*}
</div>
<p class="continuation">where <span class="process-math">\(C_n\)</span> are arbitrary constants, <span class="process-math">\(n={1},2,3,\cdots\text{.}\)</span></p>
<p id="p-407">Multiplying Eqs. <a href="" class="xref" data-knowl="./knowl/Xsol.html" title="Equation 7.7.1">(7.7.1)</a> and <code class="code-inline tex2jax_ignore">[cross-reference to target(s) "Tsol" missing or not unique]</code> together, we have</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/Xsol.html       ">
\begin{equation*}
u_n= X_n(x)T_n(t),
\end{equation*}
</div>
<p class="continuation">satisfy the PDE <code class="code-inline tex2jax_ignore">[cross-reference to target(s) "heatpde" missing or not unique]</code> and the BCs <code class="code-inline tex2jax_ignore">[cross-reference to target(s) "heatbc" missing or not unique]</code> for each <span class="process-math">\(n=1,2,\cdots\text{.}\)</span> The functions <span class="process-math">\(u_n\)</span> are sometimes called the <dfn class="terminology">fundamental solutions</dfn> to the heat conduction problem <code class="code-inline tex2jax_ignore">[cross-reference to target(s) "heatpde" missing or not unique]</code>,<code class="code-inline tex2jax_ignore">[cross-reference to target(s) "heatic" missing or not unique]</code> and <code class="code-inline tex2jax_ignore">[cross-reference to target(s) "heatbc" missing or not unique]</code>. Then the formal solution is given by</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/Xsol.html       ">
\begin{equation*}
u(x,t)=\sum_{n=1}^{\infty}u_n(x,t)=\sum_{n=1}^{\infty}C_n e^{-n^2\pi^2\alpha^2 t/L^2}\sin(n\pi x/L).
\end{equation*}
</div>
<p class="continuation">It remains only to satisfy the initial condition <code class="code-inline tex2jax_ignore">[cross-reference to target(s) "heatic" missing or not unique]</code>.</p>
<p id="p-408">To satisfy the initial condition, set <span class="process-math">\(t=0\text{,}\)</span> we get</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
u(x,0)=\sum_{n=1}^{\infty}C_n \sin\frac{n\pi x}{L}=f(x),\qquad 0\leq x\leq L.
\end{equation*}
</div>
<p class="continuation">In other words, we need to choose the coefficients <span class="process-math">\(C_n\)</span> so that the series of sine functions converges to the initial temperature distribution <span class="process-math">\(f(x)\)</span> for <span class="process-math">\(0 \leq x \leq L\text{.}\)</span> The series is just the Fourier sine series for <span class="process-math">\(f\)</span> and its coefficients are given by</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
C_n=\frac{2}{L}\int_{0}^L f(x)\sin\frac{n\pi x}{L}\textrm{d}x,\qquad n=1,2,3,\cdots
\end{equation*}
</div>
<p id="p-409">: The formal solution to the following initial and boundary value problem</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
\begin{aligned}
u_t = \alpha^2u_{xx},&amp;~&amp; 0&lt;x&lt;L,\quad t&gt;0\\
u(x,0) = f(x),&amp;~&amp; 0\leq x\leq L\\
u(0,t)=0,~~~ u(L,t)=0,&amp;~&amp; t&gt;0.
\end{aligned}
\end{equation*}
</div>
<p class="continuation">is</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
u(x,t)=\sum_{n=1}^{\infty}u_n(x,t)=\sum_{n=1}^{\infty}C_n e^{-\left(\frac{n\pi\alpha}{L}\right)^2t}\sin\dfrac{n\pi x}{L},
\end{equation*}
</div>
<p class="continuation">where</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
C_n=\frac{2}{L}\int_{0}^L f(x)\sin\frac{n\pi x}{L}\textrm{d}x,\qquad n=1,2,3,\cdots.
\end{equation*}
</div>
<ul id="p-410" class="disc">
<li id="li-59"><p id="p-411">Harmonic oscillation in <span class="process-math">\(x\text{,}\)</span> exponential decay in <span class="process-math">\(t\text{:}\)</span></p></li>
<li id="li-60"><p id="p-412">Speed of decay depending on <span class="process-math">\(\lambda_n = n\pi\alpha/L\text{.}\)</span> Faster decay for larger <span class="process-math">\(n\text{,}\)</span> meaning the high frequency components are vanished quickly. After a while, what remain in the solution are the terms with small <span class="process-math">\(n\text{.}\)</span></p></li>
<li id="li-61"><p id="p-413">The <dfn class="terminology">asymptotic</dfn> or <dfn class="terminology">steady state solution</dfn> of the problem can be obtained as <span class="process-math">\(\displaystyle\lim_{t\to\infty}u(x, t)\text{,}\)</span> which is 0 here, <span class="process-math">\(\forall x\text{.}\)</span></p></li>
</ul>
<p id="p-414">Let <span class="process-math">\(\alpha=1\)</span> and <span class="process-math">\(L=1\text{.}\)</span> If <span class="process-math">\(f(x)=10\sin \pi x\text{,}\)</span> then we have <span class="process-math">\(C_1 =10\)</span> and all other <span class="process-math">\(C_n = 0\text{,}\)</span> the solution is</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
u(x,t)=10e^{-\pi^2t}\sin \pi x.
\end{equation*}
</div>
<p class="continuation">If now let <span class="process-math">\(f(x) = 10\sin 3\pi x\text{,}\)</span> then <span class="process-math">\(C_3 = 10\)</span> and all other <span class="process-math">\(C_n = 0\text{,}\)</span> and the solution is</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
u(x,t)=10e^{-9\pi^2t}\sin 3\pi x.
\end{equation*}
</div>
<p class="continuation">If the initial temperature is</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
f(x)=10\sin \pi x + 10\sin 3\pi x,
\end{equation*}
</div>
<p class="continuation">the solution would be</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
u(x,t)=10e^{-\pi^2t}\sin \pi x +10e^{-9\pi^2t}\sin 3\pi x.
\end{equation*}
</div>
<p id="p-415">Let <span class="process-math">\(\alpha=1\text{.}\)</span> If <span class="process-math">\(f(x)= x\)</span> on <span class="process-math">\([0,L]\text{,}\)</span> then the solution is</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
u(x,t)=\sum_{n=1}^{\infty}C_n e^{-\frac{n^2\pi^2t}{L^2}}\sin\dfrac{n\pi x}{L},\qquad n=1,2,3,\cdots
\end{equation*}
</div>
<p class="continuation">where</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
C_n=\frac{2L}{n\pi}(-1)^{n+1}.
\end{equation*}
</div>
<p class="continuation">based on the result of the Fourier sine expansion in the last example from Section <code class="code-inline tex2jax_ignore">[cross-reference to target(s) "hrex1" missing or not unique]</code>.</p></section></div></main>
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